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Simplifying 6v(v + 9) = 1 Reorder the terms: 6v(9 + v) = 1 (9 * 6v + v * 6v) = 1 (54v + 6v2) = 1 Solving 54v + 6v2 = 1 Solving for variable 'v'. Reorder the terms: -1 + 54v + 6v2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 54v + 6v2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.1666666667 + 9v + v2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 9v + 0.1666666667 + v2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 9v + v2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 9v + v2 = 0 + 0.1666666667 9v + v2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 9v + v2 = 0.1666666667 The v term is 9v. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9v + 20.25 + v2 = 0.1666666667 + 20.25 Reorder the terms: 20.25 + 9v + v2 = 0.1666666667 + 20.25 Combine like terms: 0.1666666667 + 20.25 = 20.4166666667 20.25 + 9v + v2 = 20.4166666667 Factor a perfect square on the left side: (v + 4.5)(v + 4.5) = 20.4166666667 Calculate the square root of the right side: 4.518480571 Break this problem into two subproblems by setting (v + 4.5) equal to 4.518480571 and -4.518480571.Subproblem 1
v + 4.5 = 4.518480571 Simplifying v + 4.5 = 4.518480571 Reorder the terms: 4.5 + v = 4.518480571 Solving 4.5 + v = 4.518480571 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = 4.518480571 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = 4.518480571 + -4.5 v = 4.518480571 + -4.5 Combine like terms: 4.518480571 + -4.5 = 0.018480571 v = 0.018480571 Simplifying v = 0.018480571Subproblem 2
v + 4.5 = -4.518480571 Simplifying v + 4.5 = -4.518480571 Reorder the terms: 4.5 + v = -4.518480571 Solving 4.5 + v = -4.518480571 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = -4.518480571 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = -4.518480571 + -4.5 v = -4.518480571 + -4.5 Combine like terms: -4.518480571 + -4.5 = -9.018480571 v = -9.018480571 Simplifying v = -9.018480571Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.018480571, -9.018480571}
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